Question

The radius of gyration about an axis through the center of a hollow sphere with external radius a and internal radius b is

• A. $\sqrt{\frac{2}{5} \frac{\left(a^{3 }-b^{3}\right)}{\left(a^{5} -b^{5}\right)}} $
• B. $\sqrt{\frac{1}{4} \frac{\left(a^{4}-b^{4}\right)}{\left(a^{2} -b^{2}\right)}} $
• C. $\sqrt{\frac{1}{2} \frac{\left(a^{5}-b^{5}\right)}{\left(a^{3} -b^{3}\right)}} $
• D. $\sqrt{\frac{2}{5} \frac{\left(a^{5}-b^{5}\right)}{\left(a^{3} -b^{3}\right)}} $
• E. $\sqrt{\frac{5}{2} \frac{\left(a^{4}-b^{4}\right)}{\left(a^{2} -b^{2}\right)}} $

Answer 1

Answer: (D)

As we know moment of inertia of a hollow sphere with external radius $a$ and internal radius $b$ is
$I=\frac{2}{5} M\left(\frac{a^{5}-b^{5}}{a^{3}-b^{3}}\right)$
Radius of gyration, $K=\sqrt{\frac{I}{M}}$
where, $M=$ mass of sphere.
so, $ K=\frac{\sqrt{\frac{2}{5} M\left(\frac{a^{5}-b^{5}}{a^{3}-b^{3}}\right)}}{M}$
$\Rightarrow K=\sqrt{\frac{2}{5}\left(\frac{a^{5}-b^{5}}{a^{3}-b^{3}}\right)}$